12.1.1 Indeterminate Forms Flashcards
1
Q
Indeterminate Forms
A
- A limit of a function is called an indeterminate form when it produces a mathematically meaningless expression. Two types of indeterminate forms are 0/0 and ∞/∞.
- Some indeterminate forms can be solved by using algebraic tricks such as canceling or dividing by the highest power of x.
2
Q
note
A
- When taking limits, sometimes you will encounter expressions whose meanings can be interpreted in different ways. These limits are called indeterminate forms. 0/0 is one example.
- One camp says that the indeterminate form equals one because it is a number divided by itself.
- Another says that zero divided by anything is zero.
- A third says that any number divided by zero is infinity.
- Similar arguments hold for the form.
- When an indeterminate form arises, you will have to do more work.
- One algebraic trick involves factoring the numerator and the denominator.
- In this case, you can cancel the (x – 3) factors as long as you promise not to let x be equal to three.
- To evaluate this limit, look for the highest power.
- In this case, x 3 is the highest power, so divide the numerator and denominator by it. You are essentially multiplying by a form of one.
- Now all the terms have x in the denominator except one. Those terms will approach zero.
- The result is not an indeterminate form. It is negative infinity
3
Q
Evaluate limx→∞ 4x5+10x3+9x2+2x+12x5−3x4−9x+5
A
2
4
Q
Evaluate limx→3 x3−27x2−2x−3.
A
27/4
5
Q
Evaluate limx→∞ 8x8−x5+x2+11−2x8
A
-4
6
Q
Evaluate limx→100100x−x2x3−100x2.
A
-1/100
7
Q
Evaluate limx→2 x3−4x3x−6.
A
8/3
8
Q
Evaluate limx→1 x12−2x11+x10x3−x2−x+1
A
1/2
9
Q
Evaluate limx→∞ 2x3−33x3−4x2+1.
A
2/3
10
Q
Evaluate limx→3 2x−6x2−4x+3.
A
1
11
Q
Evaluate limx→∞ 2x2000−5x+4002x2001+2001
A
0
12
Q
Evaluate limx→2 x3−2x210x−20
A
2/5
13
Q
Evaluate limx→0 x3−3x211x5−4x2.
A
3/4
14
Q
Evaluate limx→1 x2+2x−3x−1
A
4
